Finite Group Databasebeta

← #27171657

Subgroups — #27171657

Order 12928.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265944C211yesmaximal, characteristic, complemented
2#20265741C211yesmaximal, characteristic
3#20265181C212yesmaximal, complemented
4#20264933C212yesmaximal, complemented
5#410×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7#15692262—24no—
8C2×#13108058—22no—
9C2×#13108058C2212yes—
10#15692251C412yescomplemented
11#15692244C412yescomplemented
12#15692229—22no—
13#15692223—22no—
14#15692214C2211yescharacteristic
15#15692175—24no—
16C2×#41×C101C2211yescharacteristic
17#117×C101C2212yes—
18#99×C101—22no—
19C23×D101—21no—
20C2×D404—22no—
21C2×D404—22no—
22C2×D404—21no—
23C2×D404C2×C412yes—
24C2×C22⋊D101—24no—
25#13108058—24no—
26#13108058—22no—
27#13108058—22no—
28C2×#11717579—24no—
29C2×#11717579D414yescomplemented
30C4⋊Dic101—21no—
31C22⋊Dic101—21no—
32#13108051—24no—
33C4×Dic101C2×C412yes—
34#41×C101—22no—
35#41×C101—22no—
36C2×Q8×C101C2×C411yescharacteristic
37C2×D4×C101C2×C411yescharacteristic
38C2×D4×C101D412yes—
39C22×C4×C101D412yes—
40C22×C4×C101C2311yescharacteristic
41C4⋊C4×C101—22no—
42C22⋊C4×C101—22no—
43C42×C101—22no—
44C23×C101C22⋊C412yes—
45C23×C101C2×D411yescharacteristic
46C22×D101—22no—
47C22×D101—42no—
48C22×D101—82no—
49Q8×C101—22no—
50D4×C101—24no—
51D4×C101—22no—
52C2×C4×C101—22no—
53C2×C4×C101—22no—
54C2×C4×C101—42no—
55C2×C4×C101C22⋊C412yes—
56C2×C4×C101C22×C411yescharacteristic
57C2×C4×C101C2×D412yes—
58C2×C4×C101C2×D411yescharacteristic
59C2×C4×C101#4112yes—
60C22⋊D101—84no—
61C2×Dic101—22no—
62D404—24no—
63D404—41no—
64D404—41no—
65#11717579—24no—
66C22×C101—22no—
67C22×C101—41no—
68C22×C101—42no—
69C22×C101C2×C22⋊C411yescharacteristic
70C22×C101#12112yes—
71C2×D101—44no—
72C2×D101—42no—
73C2×D101—84no—
74C4×C101—22no—
75C4×C101—82no—
76C4×C101C4×D412yes—
77C4×C101#12011yescharacteristic
78C4×C101#12311yescharacteristic
79Dic101—42no—
80C2×C101—21no—
81C2×C101—22no—
82C2×C101—42no—
83C2×C101#38611yescharacteristic
84D101—82no—
85#1910—1011nomaximal, Sylow
86C101#191011yescharacteristic, complemented, Sylow
87#568—1011no—
88#535—1011no—
89#418—1012no—
90#410D10111yescharacteristic, complemented
91H(3,ℤ/4)—1012no—
92C2×#41C2×D10111yescharacteristic
93C22×D4—1011no—
94#131—2024no—
95C2×M16—1012no—
96C2×M16—2022no—
97#127—1012no—
98#124—1012no—
99#121—2022no—
100#120—2022no—
101#117—1011no—
102#117C2×D10112yes—
103#102—2024no—
104#99—22no—
105C24—2021no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D10111yescharacteristic
109C2×D4—1012no—
110C2×D4—2024no—
111C2×D4—2022no—
112C2×D4—2021no—
113C2×D4—2022no—
114C2×D4C4×D10111yescharacteristic
115C2×D4C22⋊D10112yes—
116C22×C4D40412yes—
117C22×C4C22×D10111yescharacteristic
118M16—2024no—
119M16—2022no—
120M16—2022no—
121C2×C8—1014no—
122C2×C8—2024no—
123C4⋊C4—22no—
124C4⋊C4—2021no—
125C22⋊C4—22no—
126C22⋊C4—2021no—
127C22⋊C4—2024no—
128C42—22no—
129C42—1012no—
130C23—8082no—
131C23—4042no—
132C23—2022no—
133C23#1310805112yes—
134C23C2×C22⋊D10111yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—8084no—
139D4—4041no—
140D4—4041no—
141D4—2024no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—2022no—
146C2×C4#1310805112yes—
147C2×C4C2×C4×D10111yescharacteristic
148C2×C4D4×D10112yes—
149C2×C4C2×D40411yescharacteristic
150C2×C4#1310807312yes—
151C8—2024no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—8084no—
156C22—4044no—
157C22—4042no—
158C22C2×#1310805111yescharacteristic
159C22#1569222712yes—
160C4—22no—
161C4—82no—
162C4—4042no—
163C4#1569221512yes—
164C4#1569222511yescharacteristic
165C4#1569223911yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—8082no—
170C2#2026505411yescharacteristic
subgroupG/H
centerC2#20265054
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#410×C101C2
socleC2×C101#386
prime radical#20265741C2
cosocleC22×C4×C101C23
hypercenter#410D101
hypocenterC101#1910
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20265944C2semidirect
3. #20265181C2semidirect
4. #20264933C2semidirect
5. #410×C101C2semidirect
10. #15692251C4semidirect
11. #15692244C4semidirect
29. C2×#11717579D4semidirect
86. C101#1910semidirect
90. #410D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202659442
#202657412
#202651812
#202649332
#410×C1012
#1910101